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<h3 style="text-align:center;">Numerical evaluation of the second virial coefficient</h3>

<p class="header_title">Introduction</p>

<p>The pressure equation of state for a dilute gas can be written as</p>
<p class="center">
PV/Nk T = 1 + &#961;B<sub>2</sub>(T) + &#961;<sup>2</sup>
B<sub>3</sub>(T) + &#961;<sup>3</sup>
B<sub>4</sub>(T) + &#8230;,</p>
<p>
where P is the pressure, V is the volume, N is the number of particles, T is the temperature, and k is Boltzmann's constant. The density &#961; = N/V, and B<sub>n</sub> is the nth virial coefficient.
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<p>&nbsp;&nbsp;&nbsp;&nbsp; The second virial coefficient B<sub>2</sub> is given by the integral</p>
<p class="center">
<img src="b2.jpg" alt="" align="middle" >,
</p><p>
where &#946; = 1/kT and u(r) is the interparticle potential. Except for simple forms of u(r), the integral must be done numerically.</p>

<p>&nbsp;&nbsp;&nbsp;&nbsp; The program computes the integral for the Lennard-Jones potential at various temperatures. Simpson's rule is used. The program uses units such that the Lennard-Jones parameters &#949; and &#963; are set equal to unity.</p>
 

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<applet
 code="org.opensourcephysics.davidson.applets.ApplicationApplet.class"
 archive="./stp.jar" codebase="../" align="top" height="40"
 hspace="0" vspace="0" width="150"> <param name="target"
 value="org.opensourcephysics.stp.thermalcontact.ThermalContactApp"> <param name="title"
 value="Applet"> <param name="singleapp" value="true">
</applet>
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<p class="header_title">Problems</p>

<ol>

<li>Describe the qualitative temperature dependence of B<sub>2</sub>. Is it positive or negative at low/high temperatures? Is there a temperature at which B<sub>2</sub> equals zero?</li>

<li>The Lennard-Jones potential is a reasonable approximation to the interparticle potential for liquid Argon with &#949;/k = 119.5 K and &#963; = 3.76 &#215; 10<sup>-10</sup>m. Given the numerical results you found in Problem 1, what is the temperature at which B<sub>2</sub> = 0? This temperature is called the Boyle temperature.</li>

<li>Simple arguments given in standard textbooks on statistical mechanics show that B<sub>2</sub> has the approximate form, B<sub>2</sub> = b - a/kT. Interpret b and a as fit parameters and determine approximate values for and b from your results for B<sub>2</sub>.</li>


</ol>

<p class="header_title">Java Classes</p>

<ul>
<li>SecondVirialApp</li>

</ul>

<p class = "small">Updated 10 May 2008.</p>
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